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Question Number 208445    Answers: 1   Comments: 0

u_(n+1) = u_n −u_n ^3 , u_0 ∈]0,1[ v_n = (1/u_(n+1) ^2 )−(1/u_n ^2 ) = f(u_n ^2 ) ; f(x) = ((2−x)/((1−x)^2 )) v_n converges to 2, v_n is decreasing . show that v_n ≥ 2 x_n =(1/(n+1))Σ_(m=0) ^m (v_m ) . show that x_0 ≥x_n ≥v_n . show that x_n is decreasing and lim_(n→∞) x_n = l ≥2 . show that 2x_(n+1) −x_n ≤v_(n+1) and deduce l . express x_(n+1) −x_n interms of u_n . deduce lim_(n→∞) nu_n ^2

$$\left.{u}_{{n}+\mathrm{1}} \:=\:{u}_{{n}} −{u}_{{n}} ^{\mathrm{3}} ,\:{u}_{\mathrm{0}} \in\right]\mathrm{0},\mathrm{1}\left[\right. \\ $$$${v}_{{n}} \:=\:\frac{\mathrm{1}}{{u}_{{n}+\mathrm{1}} ^{\mathrm{2}} }−\frac{\mathrm{1}}{{u}_{{n}} ^{\mathrm{2}} }\:=\:{f}\left({u}_{{n}} ^{\mathrm{2}} \right)\:;\:{f}\left({x}\right)\:=\:\frac{\mathrm{2}−{x}}{\left(\mathrm{1}−{x}\right)^{\mathrm{2}} } \\ $$$${v}_{{n}} \:{converges}\:{to}\:\mathrm{2},\:{v}_{{n}} \:{is}\:{decreasing} \\ $$$$.\:{show}\:{that}\:{v}_{{n}} \:\geqslant\:\mathrm{2} \\ $$$${x}_{{n}} =\frac{\mathrm{1}}{{n}+\mathrm{1}}\underset{{m}=\mathrm{0}} {\overset{{m}} {\sum}}\left({v}_{{m}} \right) \\ $$$$.\:{show}\:{that}\:{x}_{\mathrm{0}} \geqslant{x}_{{n}} \geqslant{v}_{{n}} \\ $$$$.\:{show}\:{that}\:{x}_{{n}} \:{is}\:{decreasing}\:{and}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}{x}_{{n}} =\:{l}\:\geqslant\mathrm{2} \\ $$$$.\:{show}\:{that}\:\mathrm{2}{x}_{{n}+\mathrm{1}} −{x}_{{n}} \leqslant{v}_{{n}+\mathrm{1}} \:{and}\:{deduce}\:{l} \\ $$$$.\:{express}\:{x}_{{n}+\mathrm{1}} −{x}_{{n}} \:{interms}\:{of}\:{u}_{{n}} \\ $$$$.\:{deduce}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}{nu}_{{n}} ^{\mathrm{2}} \\ $$

Question Number 208293    Answers: 1   Comments: 0

W Z_( ∩ + (( determinant ()2)/))

Question Number 208217    Answers: 3   Comments: 0

1^2 +2^2 +3^2 +5^2 +8^2 +13^2 +21^2 =?

$$\mathrm{1}^{\mathrm{2}} +\mathrm{2}^{\mathrm{2}} +\mathrm{3}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} +\mathrm{8}^{\mathrm{2}} +\mathrm{13}^{\mathrm{2}} +\mathrm{21}^{\mathrm{2}} =? \\ $$

Question Number 207979    Answers: 1   Comments: 3

generate nth term for the sequence: 1, 1, 1, 2, 3, 5, 9, 18, 35, 75

generate nth term for the sequence: 1, 1, 1, 2, 3, 5, 9, 18, 35, 75

Question Number 207731    Answers: 2   Comments: 0

$$\:\:\:\downharpoonleft\underline{\:} \\ $$

Question Number 207594    Answers: 2   Comments: 0

((xcosθ)/a) + ((ysinθ)/b) = 1 xsinθ − ycosθ = (√(a^2 sin^2 θ + b^2 cos^2 θ)) Eliminate θ.

$$\frac{{x}\mathrm{cos}\theta}{{a}}\:+\:\frac{{y}\mathrm{sin}\theta}{{b}}\:=\:\mathrm{1} \\ $$$${x}\mathrm{sin}\theta\:−\:{y}\mathrm{cos}\theta\:=\:\sqrt{{a}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \theta\:+\:{b}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \theta} \\ $$$$\mathrm{Eliminate}\:\theta. \\ $$

Question Number 207546    Answers: 1   Comments: 0

The real roots of the equation x^2 +6x+c=0 differ by 2n, where n is a real non−zero. Show that n^2 =9−c Given that the roots also have opposite signs, find the set of possible values of n

$$\mathrm{The}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{x}^{\mathrm{2}} +\mathrm{6x}+\mathrm{c}=\mathrm{0} \\ $$$$\mathrm{differ}\:\mathrm{by}\:\mathrm{2n},\:\mathrm{where}\:\mathrm{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{non}−\mathrm{zero}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{n}^{\mathrm{2}} =\mathrm{9}−\mathrm{c} \\ $$$$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{also}\:\mathrm{have}\:\mathrm{opposite} \\ $$$$\mathrm{signs},\:\mathrm{find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n} \\ $$

Question Number 207416    Answers: 2   Comments: 0

$$\:\:\: \\ $$

Question Number 207194    Answers: 1   Comments: 0

Let x = cos(π/9) Show that 8x^3 −6x−1=0 Deduce x is not rational

$${Let}\:\:{x}\:=\:{cos}\frac{\pi}{\mathrm{9}}\: \\ $$$${Show}\:{that}\:\mathrm{8}{x}^{\mathrm{3}} −\mathrm{6}{x}−\mathrm{1}=\mathrm{0} \\ $$$${Deduce}\:{x}\:{is}\:{not}\:\:{rational}\: \\ $$

Question Number 207042    Answers: 3   Comments: 0

Prove that the sum of a square function is ((n(n+1)(2n+1))/6)

$${Prove}\:{that}\:{the}\:{sum}\:{of}\:{a}\:{square}\:{function} \\ $$$${is}\:\frac{{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}{\mathrm{6}} \\ $$

Question Number 206919    Answers: 2   Comments: 0

Question Number 206924    Answers: 2   Comments: 1

Question Number 206848    Answers: 1   Comments: 1

Question Number 206805    Answers: 2   Comments: 0

Question Number 206783    Answers: 1   Comments: 1

Question Number 206709    Answers: 3   Comments: 4

Find the missing number determinant ((( 72),(24),( 6)),(( 96),(16),(12)),((108),( ?),(18))) A.12 B.16 C.18 D.20 Please help...

$$\:\mathrm{Find}\:\mathrm{the}\:\mathrm{missing}\:\mathrm{number} \\ $$$$\:\:\:\:\:\:\:\:\:\:\begin{array}{|c|c|c|}{\:\mathrm{72}}&\hline{\mathrm{24}}&\hline{\:\:\mathrm{6}}\\{\:\mathrm{96}}&\hline{\mathrm{16}}&\hline{\mathrm{12}}\\{\mathrm{108}}&\hline{\:?}&\hline{\mathrm{18}}\\\hline\end{array} \\ $$$$\mathrm{A}.\mathrm{12}\:\:\:\:\:\:\mathrm{B}.\mathrm{16}\:\:\:\:\mathrm{C}.\mathrm{18}\:\:\:\:\:\:\mathrm{D}.\mathrm{20} \\ $$$$\mathrm{Please}\:\mathrm{help}... \\ $$

Question Number 206675    Answers: 1   Comments: 0

There are three positive integers a, b, and c such that their average is 35 and a ≤ b ≤ c. If the median is (a + 18), then find the minimum possible value of c. (1) 41. (2) 42. (3) 39. (4) 40

$$ \\ $$There are three positive integers a, b, and c such that their average is 35 and a ≤ b ≤ c. If the median is (a + 18), then find the minimum possible value of c. (1) 41. (2) 42. (3) 39. (4) 40

Question Number 206636    Answers: 0   Comments: 1

resoudre dans N^2 l equation 2x^2 +3y^2 =35

$${resoudre}\:{dans}\:{N}^{\mathrm{2}} \:\:{l}\:{equation} \\ $$$$\mathrm{2}{x}^{\mathrm{2}} +\mathrm{3}{y}^{\mathrm{2}} =\mathrm{35} \\ $$

Question Number 206608    Answers: 1   Comments: 0

Question Number 206399    Answers: 2   Comments: 1

Question Number 206353    Answers: 0   Comments: 6

Question Number 206309    Answers: 2   Comments: 0

Find the ways to express 11025 as product of two factors. (a) 13 (b) 14 (c) 26 (d) 27

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{ways}\:\mathrm{to}\:\mathrm{express}\:\mathrm{11025}\: \\ $$$$\mathrm{as}\:\mathrm{product}\:\mathrm{of}\:\mathrm{two}\:\mathrm{factors}. \\ $$$$\left(\mathrm{a}\right)\:\mathrm{13}\:\:\:\:\:\:\:\:\:\left(\mathrm{b}\right)\:\mathrm{14}\:\:\:\:\:\:\:\:\left(\mathrm{c}\right)\:\mathrm{26}\:\:\:\:\:\:\:\:\:\:\:\left(\mathrm{d}\right)\:\mathrm{27} \\ $$

Question Number 206253    Answers: 2   Comments: 0

f(x)= log_( 2) ( x + 2(√x) +4 ) ⇒ f^( −1) ( 13 −4(√3) ) = ? −−−−−

$$\:\:\: \\ $$$$\:\:\:\:\:\:\:{f}\left({x}\right)=\:{log}_{\:\mathrm{2}} \:\left(\:{x}\:+\:\mathrm{2}\sqrt{{x}}\:+\mathrm{4}\:\right) \\ $$$$\:\:\:\:\:\:\:\Rightarrow\:\:{f}^{\:−\mathrm{1}} \left(\:\mathrm{13}\:−\mathrm{4}\sqrt{\mathrm{3}}\:\right)\:=\:? \\ $$$$\:\:\:\:\:\:\:−−−−− \\ $$$$\:\:\:\: \\ $$

Question Number 206230    Answers: 1   Comments: 0

Expand x^2 + 2x + 3 respect to x = −2. (a) (x − 2)^2 −2(x + 2) + 3 (b) (x + 2)^2 −2(x + 2) + 3 (c) (x + 2)^2 + 2(x + 2) + 3 (d) (x − 2)^2 −2(x − 2) − 3 is it taylors theorem?

$$\mathrm{Expand}\:\:\:\:{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:\mathrm{3}\:\:\:{respect}\:{to}\:{x}\:=\:−\mathrm{2}. \\ $$$$\left(\mathrm{a}\right)\:\left({x}\:−\:\mathrm{2}\right)^{\mathrm{2}} −\mathrm{2}\left({x}\:+\:\mathrm{2}\right)\:+\:\mathrm{3} \\ $$$$\left(\mathrm{b}\right)\:\left({x}\:+\:\mathrm{2}\right)^{\mathrm{2}} −\mathrm{2}\left({x}\:+\:\mathrm{2}\right)\:+\:\mathrm{3} \\ $$$$\left(\mathrm{c}\right)\:\left({x}\:+\:\mathrm{2}\right)^{\mathrm{2}} +\:\mathrm{2}\left({x}\:+\:\mathrm{2}\right)\:+\:\mathrm{3} \\ $$$$\left(\mathrm{d}\right)\:\left({x}\:−\:\mathrm{2}\right)^{\mathrm{2}} −\mathrm{2}\left({x}\:−\:\mathrm{2}\right)\:−\:\mathrm{3} \\ $$$$ \\ $$$$\mathrm{is}\:\mathrm{it}\:\mathrm{taylors}\:\mathrm{theorem}? \\ $$

Question Number 206038    Answers: 2   Comments: 0

(√(1 + 2023(√(1 + 2024(√(1+ 2025(√(1 + 2026(√(1 + ..............∞)))))))))) = ?

$$\sqrt{\mathrm{1}\:+\:\mathrm{2023}\sqrt{\mathrm{1}\:+\:\mathrm{2024}\sqrt{\mathrm{1}+\:\mathrm{2025}\sqrt{\mathrm{1}\:+\:\mathrm{2026}\sqrt{\mathrm{1}\:+\:..............\infty}}}}}\:=\:?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 205893    Answers: 0   Comments: 4

Question Number 205842    Answers: 2   Comments: 0

((32^(32^(32) ) )/9) ≡^R ?

$$\frac{\mathrm{32}^{\mathrm{32}^{\mathrm{32}} } }{\mathrm{9}}\:\overset{\mathrm{R}} {\equiv}\:? \\ $$

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